n^2+n=22

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Solution for n^2+n=22 equation:


Simplifying
n2 + n = 22

Reorder the terms:
n + n2 = 22

Solving
n + n2 = 22

Solving for variable 'n'.

Reorder the terms:
-22 + n + n2 = 22 + -22

Combine like terms: 22 + -22 = 0
-22 + n + n2 = 0

Begin completing the square.

Move the constant term to the right:

Add '22' to each side of the equation.
-22 + n + 22 + n2 = 0 + 22

Reorder the terms:
-22 + 22 + n + n2 = 0 + 22

Combine like terms: -22 + 22 = 0
0 + n + n2 = 0 + 22
n + n2 = 0 + 22

Combine like terms: 0 + 22 = 22
n + n2 = 22

The n term is n.  Take half its coefficient (0.5).
Square it (0.25) and add it to both sides.

Add '0.25' to each side of the equation.
n + 0.25 + n2 = 22 + 0.25

Reorder the terms:
0.25 + n + n2 = 22 + 0.25

Combine like terms: 22 + 0.25 = 22.25
0.25 + n + n2 = 22.25

Factor a perfect square on the left side:
(n + 0.5)(n + 0.5) = 22.25

Calculate the square root of the right side: 4.716990566

Break this problem into two subproblems by setting 
(n + 0.5) equal to 4.716990566 and -4.716990566.

Subproblem 1

n + 0.5 = 4.716990566 Simplifying n + 0.5 = 4.716990566 Reorder the terms: 0.5 + n = 4.716990566 Solving 0.5 + n = 4.716990566 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = 4.716990566 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = 4.716990566 + -0.5 n = 4.716990566 + -0.5 Combine like terms: 4.716990566 + -0.5 = 4.216990566 n = 4.216990566 Simplifying n = 4.216990566

Subproblem 2

n + 0.5 = -4.716990566 Simplifying n + 0.5 = -4.716990566 Reorder the terms: 0.5 + n = -4.716990566 Solving 0.5 + n = -4.716990566 Solving for variable 'n'. Move all terms containing n to the left, all other terms to the right. Add '-0.5' to each side of the equation. 0.5 + -0.5 + n = -4.716990566 + -0.5 Combine like terms: 0.5 + -0.5 = 0.0 0.0 + n = -4.716990566 + -0.5 n = -4.716990566 + -0.5 Combine like terms: -4.716990566 + -0.5 = -5.216990566 n = -5.216990566 Simplifying n = -5.216990566

Solution

The solution to the problem is based on the solutions from the subproblems. n = {4.216990566, -5.216990566}

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